44 2 «Æ Vol.44, No.2 2015 3 ADVANCES IN MATHEMATICS(CHINA) Mar., 2015 doi: 10.11845/sxjz.2015056b Viscosity approximation methods for equilibrium problems and a finite family of nonspreading mappings in a Hilbert space HUO Xiaoyan, ZHOU Haiyun, HE Jiangyan (Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, 050003, P. R. China) Abstract: In this paper, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonspreading mappings in a real Hilbert space. We obtain a strong convergence theorem for the sequences generated by this iterative scheme. Keywords: viscosity approximation method; equilibrium problem; fixed point; nonspreading mapping MR(2010) Subject Classification: 47B38; 47G10 / CLC number: O177.91 Document code: A Article ID: 1000-0917 0 Introduction Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let F be a bifunction of C C into R, where R is the set of real numbers. The equilibrium problem for F: C C R is to find x C such that F(x,y) 0 for all y C. (0.1) The set of solutions of (0.1) is denoted by EP(F). Given a mapping T : C H, let F(x,y) = Tx,y x for all x,y C. Then z EP(F) if and only if Tz,y z 0 for all y C, i.e., z is a solution of the variational inequality. For solving the equilibrium problem for a bifunction F: C C R, let us assume that F satisfies the following conditions: (A1) F(x,x) = 0 for all x C; (A2) F is monotone, i.e., F(x,y)+F(y,x) 0 for all x,y C; (A3) For each x,y,z C, lim t 0 F(tz +(1 t)x,y) F(x,y); (A4) For each x C, y F(x,y) is convex and lower semicontinous. A mapping S : C C is called nonexpansive if Received date: 2015-03-09. Foundation item: Supported by NSFC (No. 11071053) E-mail: huoxiaoyan2006@163.com Sx Sy x y for all x,y C.
288 Å 44 A point x C is called a fixed point of S if Sx = x. We denote by F(S) the set of fixed points of S. If C H is bounded, closed and convex and S is a nonexpansive mapping of C into itself, then F(S) is nonempty. A mapping S : C C is called quasi-nonexpansive if Sx z x z for all x,y C,z Fix(S). A mapping S : C C is called nonspreading if 2 Sx Sy 2 Sx y 2 + x Sy 2 for all x,y C (0.2) or Sx Sy 2 x y 2 +2 x Sx,y Sy for all x,y C. (0.3) Remark 1 When F(S), a nonspreading mapping is a quasi-nonexpansive mapping. It is clear that every nonexpansive mapping with a nonempty set of fixed points is quasi-nonexpansive. Let {T i } N i=1 be a finite family of nonspreading mappings. Assume throughout the rest of the paper that N Fix(T i ). i=1 For n > N,T n is understood as T nmodn with the mod function taking values in {1,2,,N}, see [1]. Let λ n,1,λ n,2, λ n,n (0,1],n N. Given the mappings T 1,T 2,,T N, following [2] one can define, for each n, mappings U n,1,u n,2,,u n,n by U n,1 = λ n,1 T 1 +(1 λ n,1 I), U n,2 = λ n,2 T 2 U n,1 +(1 λ n,2 I), U n,n 1 = λ n,n 1 T N 1 U n,n 2 +(1 λ n,n 1 I), W n U n,n = λ n,n T N U n,n 1 +(1 λ n,n I). (0.4) Such a mapping W n is called the W n mapping generated by T 1,T 2,,T N and λ n,1,λ n,2, λ n,n. In 2010, Paul-Emile Maingé [3] proposed a new analysis of viscosity approximation method in some framework which takes into account the wide class of demicontractive operators and established the strong convergences of the sequence given by x n+1 = α n Cx n +(1 α n )T ω x n, where {α n } is a slow vanishing sequence, ω (0,1],T ω := (1 ω)i+ωt, with some conditions on T. And at the same time, Kurokawa and Takahashi [4] proved the following strong convergence theorem for nonspreading mappings in Hilbert space.
2,, : Viscosityapproximationmethodsforequilibriumproblemsandafinitefamilyofnonspreadingmappingsi Theorem 0.1 [4] Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonspreading mapping of C into itself. Let u C and define two sequences {x n } and {z n } in C as follows: x 1 = x C and x n+1 = α n u+(1 α n )z n, z n = 1 n 1 T k x n n n=0 for all n = 1,2,, where 0 α n 1,α n 0, and n=1 α n =. If F(T) is nonempty, then {x n } and {z n } converge strongly to Pu, where P is the metric projection of H onto F(T). On the other hand, given any r > 0, it is shown [5] that under suitable hypotheses on F (to be stated precisely in Section 2), the mapping T r : H C defined by T r (x) = {z C : F(z,y)+ 1 y z,z x 0, y C} (1.5) r is single-valued and firmly nonexpansive and satisfies Fix(T r ) = EP(F). Using this result, Takahashi [6] obtain the following result for generalized hybrid mappings of C into H with equilibrium problem in a Hilbert space. Theorem 0.2 [6] Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let S be a generalized hybrid mapping of C into H. Let F be a bifunction of C C into R satisfying (A1) (A4). Let 0 < k < 1, and let g be a k contraction of H into itself. Let V be a γ strongly monotone and L Lipschitzian continuous of H into itself with γ > 0 and L > 0. Take µ,γ R as follows: 0 < µ < 2 γ L 2, 0 < γ < γ L 2 µ 2. k Suppose that F(S) EP(F). Let x 1 = x H, and let {x n } H be a sequence generated by F(u n,y)+ 1 y u n,u n x n 0, y C, r n x n+1 = β n x n +(1 β n ){α n γg(x n )+(I α n V)Su n } for all n N, where {βn} (0,1),{α n } (0,1) and {r n } (0, ) satisfy lim α n = 0; n=1 α n = ; liminf r n > 0, 0 < liminf β n limsup β n < 1. Thenthesequence{x n }convergesstronglytoz 0 F(S) EP(F), wherez 0 = P F(S) EP(F) (I V +γg)z 0. Using (0.5), V.Colao, G.marino and H.Xu [1] introduced a viscosity approximation method for finding a common element of EP(F) and Fix(W n ), where is defined by (0.4) for a finite family of nonexpansive mappings. Starting with an arbitrary element x 1 H, they defined the sequences {u n } and {x n } recursively by F(u n,y)+ 1 y u n,u n x n 0, y H, r n x n+1 = ǫ n γf(x n )+βx n +((1 β)i ǫ n A)W n u n.
290 Å 44 They proved that under certain appropriate hypotheses that both sequences {u n } and {x n } convergestronglyto a point x F which is anequilibrium point for F and isthe unique solution of the variational inequality (A γf)x,x x 0, x F EP(F). Finding an optimal point in the intersection N i=1 Fix(T i) of the fixed points set of a finite family of nonexpansive mappings with equilibrium problem in a Hilbert space is a frequently problem interest in various branches of sciences;see[7-9]. In this paper, we propose a new iteration algorithm and study viscosity approximation methods for equilibrium problems and a finite family of nonspreading mappings in a Hilbert space and then prove a strong convergence theorem. 1 Preliminaries LetH bearealhilbertspacewithinnerproduct, andnorm.when{x n }isasequence in H, x n x means that {x n } convergesweakly to x and x n x means the strong convergence. For a bounded sequence {x n } H, w w (x n ) = {x H : {x nj } {x n },s.t.x nj x} denotes the weak w limit set of {x n }. In a Hilbert space, the following equalities are well known: λx+(1 λ)y 2 = λ x 2 +(1 λ) y 2 λ(1 λ) x y 2 (1.1) 2 x y,z w = x w 2 + y z 2 x z 2 y w 2 (1.2) x,y = 1 2 x y 2 + 1 2 x 2 + 1 2 y 2 (1.3) for all x,y,z,w H and λ R. Let C be a nonempty closed convex subset of H, then for any x H, there exists a unique nearest point in C, denoted by P C (x), such that x P C (x) x y for all y C. (1.4) Such a P C is called the metric projection of H onto C. We know that P C is nonexpansive. Further, for x H and z C, z = P C (x) x z,z y 0 for all y C. We also know that for any sequence {x n } H with x n x, the following inequality holds: x n x < liminf x n y for every y H with x y. Lemma 1.1 Let {Γ n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {Γ nj } j 0 of {Γ n } which satisfies Γ nj < Γ nj+1 for all j 0. Also consider the sequence of integers {τ(n)} n n0 defined by τ(n) = max{k n Γ k < Γ k+1 }.
2,, : Viscosityapproximationmethodsforequilibriumproblemsandafinitefamilyofnonspreadingmappingsi Then {τ(n)} n n0 is a nondecreasing sequence verifying lim τ(n) = and for all n n 0, it holds that Γ τ(n) Γ τ(n)+1 and we have Γ n Γ τ(n)+1. Lemma 1.2 [5] Let C be a nonempty closed convex subset of Hilbert space H and F be a bifunction of C C R satisfying (A1) (A4), then for any given r > 0 and x H, there exists z C such that F(z,y)+ 1 y z,z x 0 for all y C. r Lemma 1.3 [5] Assume that F : C C R satisfies (A1) (A4), for r > 0 and x H, define a mapping T r : H C as follows: T r (x) = {z C : F(z,y)+ 1 y z,z x 0, y C} r for all x H. Then the following holds: (1) T r is single-valued; (2) T r is firmly nonexpansive, i.e., for any x,y H, T r x T r y 2 T r x T r y,x y. This implies that T r x T r y x y, x,y H, i.e., T r is a nonexpansive mapping; (3) F(T r ) = EP(F), r > 0; (4) EP(F) is closed and convex. Lemma 1.4 [10] Let C be a nonempty closed convex subset of Hilbert space H, T : C C a nonspreading mapping and F(T), then I T is demiclosed at θ, i.e., for any sequence {x n } C, such that x n z and x n Tx n 0, then z F(T). Lemma 1.5 [11] Let C be a nonempty closed convex subset of Hilbert space H,T : C C a quasi-nonexpansive mapping, then the fixed points F(T) of T is a nonempty closed convex subset of C. Lemma 1.6 [7] For all x,y H, there holds the inequality x+y 2 x 2 +2 y,x+y. Lemma 1.7 Let C and Ω be two nonempty closed convex subsets of a real Hilbert space H such that Ω C. Let z be a solution of the following variational inequality (I f)x,y x 0, y Ω. (VI) Let {x n } be a bounded sequence in C. If w w (x n ) Ω, then (I f)z,x n z 0. Proof Since C is a nonempty closed convex subset and {x n } is a bounded sequence in C, we know that {x nk } {x n } such that (I f)z,x n z = lim k (I f)z,x n k z. (1.5) Without loss of generalization, we can assume that x nk ν(k ), then ν Ω. By (VI) and (1.5) which obviously leads to (I f)z,x n z = (I f)z,ν z 0,
292 Å 44 that is the desired result. Lemma 2.8 Let W n be defined by (0.4), T i : i = 1,2,,N be a finite family of nonspreading mappings, if N i=1 F(T i), then we have (1) N i=1 F(T i) = F(W n ); (2)For n N,W n is quasinonexpansive; (3)For n N,W n is demi-closed. Proof (1) First show that N i=1 F(T i) F(W n ). When x N i=1 F(T i), we have T i x = x,(i = 1,2,,N), then U n,1 x = λ n,1 (T 1 x x)+x = x, U n,2 x = λ n,2 (T 2 U n,1 x x)+x = λ n,2 (T 2 x x)+x = x. Suppose when i = N 1, we have U n,n 1 x = x, then when i = N, we obtain W n x = U n,n x = λ n,n (T N U n,n 1 x x)+x = λ n,n (T N x x)+x = x, so we have x F(W n ). Next we show when F(W n ) N i=1 F(T i). When N = 1, U n,1 x = λ n,1 (T 1 x x) + x = x, then T 1 x = x. Suppose when n = N 1, T N 1 x = x, then when n = N, we have W n x = U n,n x = λ n,n (T N U n,n 1 x x)+x = λ n,n (T N x x)+x = x, then T N x = x. So x N i=1 F(T i), this completes the proof. (2)For q N i=1 F(T i), by Remark 1, when N = 1, we have U n,1 x q = λ n,1 T 1 x+(1 λ n,1 )x q = λ n,1 (T 1 x q)+(1 λ n,1 )(x q) λ n,1 T 1 x q +(1 λ n,1 ) x q λ n,1 x q +(1 λ n,1 ) x q = x q. So U n,1 is quasi-nonexpansive. Suppose U n,n 1 is quasi-nonexpansive, i.e., U n,n 1 x q x q. next we show U n,n is quasi-nonexpansive. W n x q = U n,n x q = λ n,n T N U n,n 1 x+(1 λ n,n )x q = λ n,n (T N U n,n 1 x q)+(1 λ n,n )(x q) λ n,n U n,n 1 x q +(1 λ n,n ) x q λ n,n x q) +(1 λ n,1 ) x q = x q. So W n is quasi-nonexpansive. (3)We only to show when x n x,w n x n x n 0, then x F(W n ), i.e., x N i=1 F(T i). We prove it by induction. When N = 1, U n,1 x n x n = λ n,1 (T 1 x n x n ) 0,
2,, : Viscosityapproximationmethodsforequilibriumproblemsandafinitefamilyofnonspreadingmappingsi T 1 is a nonspreadingmapping, so x Fix(T 1 ). Because of x n x, so U n,1 x n x. When N = 2, U n,2 x n x n = λ n,2 (T 2 U n,1 x n x n ) 0 T 2 U n,1 x n x n 0, T 2 U n,1 x n U n,1 x n T 2 U n,1 x n x n + U n,1 x n x n 0, so x Fix(T 2 ), i.e., U n,2 is demi-closed. Suppose U n,n 1 is demi-closed, next we show W n is demi-closed. W n x n x n = λ n,n (T N U n,n 1 x n x n ) 0, T N U n,n 1 x n U n,n 1 x n T N U n,n 1 x n x n + U n,n 1 x n x n 0, because of U n,n 1 x n x, we have x Fix(T N ), that is to say W n is demi-closed. 2 Main Results Theorem 2.1 Let C be a nonempty closed convex subset of Hilbert space H, T i : C C,i = 1,2,,N beafinitefamilyofnonspreadingmappingssuchthat N i=1 F(T i), f : C C be a contraction on C with a constant ρ (0,1), and let λ n,1,λ n,2, λ n,n be a sequence in [a,b] with 0 < a b < 1, for every n N,W n be the mapping generated by T 1,T 2,,T N and λ n,1,λ n,2, λ n,n. Let F be a bifunction from C C R satisfying (A1) (A4). Assume that N i=1 F(T i) EP(F). Let x 1 H, {x n } and {u n } be sequences defined by F(u n,y)+ 1 y u n,u n x n 0, y C r n x n+1 = β n x n +(1 β n )[α n f(x n )+(1 α n )W n u n ], where {α n } [0,1] and {r n } (0, ) satisfying (i) lim α n = 0; (ii) n=1 α n = ; (iii) 0 < liminf β n limsup β n < 1. Then both {x n } and {u n } converge strongly to z N i=1 F(T i) EP(F), where z = P N i=1 F(T i) EP(F)f(z). EP(F). Proof We split the proof into five steps. (2.1) Step 1. Show that the variational inequality (VI) has a unique solution in N i=1 F(T i) Since T i : C C (i = 1,2,,N) is a nonspreading mapping with N i=1 F(T i), by Remark 1, we know that T i is a quasi-nonexpansive mapping, and hence it follows from Lemma 1.5that N i=1 F(T i)isanonemptyclosedconvexsubsetofc. ByLemma1.3,weknowthatEP(F) is also closed convex. Consequently, N i=1 F(T i) EP(F) is nonempty closed convex, and then the metric projection P N i=1 F(T i) EP(F) is well defined. Note that z is a solution of variational inequality(vi)isequaivalenttothatz isafixedpointofp N i=1 F(T i) EP(F)f. Sincef : C C isa contraction and P N i=1 F(T i) EP(F) : C C is nonexpansive, we have P N i=1 F(T i) EP(F)f : C C is a contraction, by the Banach contraction mapping principle, we know that P N i=1 F(T i) EP(F)f has a unique fixed point z in N i=1 F(T i) EP(F), which implies that (I f)z,y z 0, y N i=1f(t i ) EP(F).
294 Å 44 Step 2. Show that {x n } is bounded. Let v N i=1 F(T 1 i) EP(F),M = max{ x 1 v, f(v) v }, then from (0.3) and 1 ρ Lemma 1.3, Lemma 2.8, noting that u n = T rn x n, we have u n v = T rn x n T rn v x n v, for all n N, (2.2) W n u n v 2 u n v 2 x n v 2. (2.3) Then by (2.1),(2.2) and (3.3) we have x n+1 v = β n x n +(1 β n )[α n f(x n )+(1 α n )W n u n ] v = β n (x n v)+(1 β n )α n (f(x n ) v)+(1 β n )(1 α n )(W n u n v) β n x n v +(1 β n )α n ρ x n v +(1 β n )α n f(v) v +(1 β n )(1 α n ) W n u n v 1 (1 (1 β n )(1 ρ)α n ) x n v +(1 β n )α n (1 ρ) 1 ρ f(v) v M. So we have x n v M for any n N and hence {x n } is bounded. We also obtain that {u n },{W n u n },{W n x n } and {f(x n )} are all bounded. Step 3. Show that W n u n u n 0. First show that W n u n x n 0. Let z be the solution of (VI), by the definition of {x n } we have x n+1 x n +(1 β n )α n (x n f(x n )) = (1 β n )(1 α n )(W n u n x n ), (2.4) so x n+1 x n +(1 β n )α n (x n f(x n )),x n z = (1 β n )(1 α n ) x n W n u n,x n z. (2.5) From (1.2) and (2.3) we have 2 x n W n u n,x n z = x n z 2 + W n u n x n 2 W n u n z 2 W n u n x n 2, (2.6) together with (2.5) and (2.6) we have x n x n+1,x n z (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2. (2.7) Furthermore, using (1.3) we have x n x n+1,x n z = 1 2 x n+1 z 2 1 2 x n x n+1 2 1 2 x n z 2 (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2, Denote Γ n = 1 2 x n z 2. Then Γ n+1 Γ n 1 2 x n x n+1 2 (1 β n )α n (I f)x n,x n z 1 2 (1 β n)(1 α n ) W n u n x n 2. (2.8)
2,, : Viscosityapproximationmethodsforequilibriumproblemsandafinitefamilyofnonspreadingmappingsi Noting that x n+1 x n = (1 β n )α n (f(x n ) x n )+(1 β n )(1 α n )(W n u n x n ) (1 β n )(α n f(x n ) x n +(1 α n ) W n u n x n ), we have that x n+1 x n 2 (1 β n ) 2 [α n f(x n ) x n +(1 α n ) W n u n x n ] 2 = (1 β n ) 2 α 2 n f(x n) x n 2 +(1 β n ) 2 (1 α n ) 2 W n u n x n 2 +2(1 β n ) 2 α n (1 α n ) f(x n ) x n W n u n x n, by (2.8) we obtain Γ n+1 Γ n + 1 2 (1 β n)(1 α n )(1 (1 β n )(1 α n )) W n u n x n 2 (1 β n )α n ( 1 2 (1 β n)α n f(x n ) x n 2 +(1 α n )(1 β n ) f(x n ) x n W n u n x n (I f)x n,x n z ). Then we will divide the proof into two cases: (2.9) Case 1: Suppose 0 Γ n+1 Γ n,n 1, i.e., {Γ n } is nonincreasing. In this case, {Γ n } is convergent, so that lim (Γ n+1 Γ n ) = 0. By using conditions (i), (iii) and the boundedness of {x n }, we obtain lim W nu n x n = 0. (2.10) Next we show that x n u n 0. Let v F(W n ) EP(F), from (1.2) we have u n v 2 = T rn x n T rn v 2 T rn x n T rn v,x n v = u n v,x n v = 1 2 ( u n v 2 + x n v 2 x n u n 2 ), i.e., u n v 2 x n v 2 x n u n 2, (2.11) together with (2.3) we have W n u n v 2 u n v 2 x n v 2 x n u n 2. (2.12) Because of the boundedness of {x n } and {W n u n }, we put M = sup n 1 { x n v + W n u n v }, then by (2.10) we obtain x n u n 2 x n v 2 W n u n v 2 M W n u n x n, (2.13) lim x n u n = 0. (2.14) Noting that W n u n u n W n u n x n + x n u n, we also have lim W nu n u n = 0. (2.15)
296 Å 44 Case 2 Supposethereexistsasubsequence{Γ nk } k 0 {Γ n },suchthatγ nk < Γ nk +1, k 0. In this case, we define τ : N N, by τ(n) = max{k n Γ k < Γ k+1 }, then it follows that Γ τ(n)+1 Γ τ(n) 0, by Lemma 1.1 and (2.9) amounts to 1 2 (1 β τ(n))(1 α τ(n) )(1 (1 β τ(n) )(1 α τ(n) )) W n u τ(n) x τ(n) 2 (1 β τ(n) )α τ(n) ((1 β τ(n) )α τ(n) f(x τ(n) ) x τ(n) 2 +(1 α τ(n) )(1 β τn ) f(x τ(n) ) x τ(n) W n u τ(n) x τ(n) (I f)x τ(n),x τ(n) z ). (2.16) Hence, by the boundedness of {x n }, conditions (i) and (iii), we immediately obtain lim W nu τ(n) x τ(n) 2 = 0, lim Γ τ(n) Γ τ(n)+1 = 0. (2.17) With the similar argument of (2.11) (2.14), we have then lim x τ(n) u τ(n) = 0, (2.18) lim W nu τ(n) u τ(n) = 0. (2.19) Step 4 Assume that there exists a subsequence {x nk } of {x n } which converges weakly to w, w w w (x n ), show that w w (x n ) F(W n ) EP(F). By Lemma 2.8,we know W n is demi-closed, so w F(W n ). Next we show w EP(F). By u n = T rn x n, we have From the monotonicity of F, we have and hence F(u n,y)+ 1 r n y u n,u n x n 0, y C. 1 r n y u n,u n x n F(u n,y) F(y,u n ), y C y u nk, 1 r nk (u nk x nk ) F(y,u nk ), y C. Since liminf r n a > 0, we have u n k x nk r nk 0 and u nk w, from (A4) we have F(y,w) 0, y C. For 0 t 1 and y C, let y t = ty + (1 t)w. Since y,w C, we have y t C and hence F(y t,w) 0. So from (A1) and (A4) we have 0 = F(y t,ty +(1 t)w) tf(y t,y)+(1 t)f(y t,w) tf(y t,y) and hence F(y t,y) 0, from (A3) we have F(w,y) 0
2,, : Viscosityapproximationmethodsforequilibriumproblemsandafinitefamilyofnonspreadingmappingsi for all y C and hence w EP(F). Therefore w F(W n ) EP(F), and w w (x n ) F(W n ) EP(F). Step 5 Finally, we show that both {x n } and {u n } converge strongly to z, where z = P F(Wn) EP(F)f(z). In the case 1, from (2.9) and (2.10), we obtain (1 β n )α n [ 1 2 (1 β n)α n f(x n ) x n 2 (I f)x n,x n z ] Γ n Γ n+1, then by condition (ii), we deduce that ( 1 2 (1 β n)α n f(x n ) x n 2 + (I f)x n,x n z ) 0 or equivalently ( as (1 β n )α n f(x n ) x n 2 0) Put F = I f, then Fx Fy,x y (1 ρ) x y 2, so which by (2.20) entails (I f)x n,x n z 0. (2.20) (I f)x n (I f)z,x n z (1 ρ) x n z 2 = 2(1 ρ)γ n, (2(1 ρ)γ n + (I f)z,x n z ) 0. Invoking Lemma 1.7, steps 1, 4 and recalling that lim Γ n exists, we have 0 2(1 ρ) lim Γ n liminf (I f)z,x n z 0, so lim Γ n = 0, i.e., {x n } converges strongly to z. In the case 2, similarly, from (2.16) we can deduce Invoking Lemma 1.7, steps 1, 4, we have so 2(1 ρ)limsupγ τ(n) liminf (I f)z,x τ(n) z. (I f)z,x τ(n) z 0, lim Γ τ(n) = 0, and lim Γ τ(n)+1 = 0 by (2.17), recalling that Γ τ(n)+1 Γ n (by Lemma 1.1), we get lim Γ n = 0, so that x n z strongly. As direct consequence of Theorem 3.1, we obtain the following result. Corollary 2.1 Let C be a nonempty closed convex subset of Hilbert space H, S : C C a nonspreading mapping such that F(S), f : C C a contraction on C with a constant ρ (0,1). Assume that F(S). Let x 1 C, {x n } be sequences defined by x n+1 = β n x n +(1 β n )[α n f(x n )+(1 α n )Sx n ],
298 Å 44 where {α n } [0,1] satisfying (i) lim α n = 0; (ii) n=1 α n = ; (iii) 0 < liminf β n limsup β n < 1. Then {x n } converge strongly to z F(S), where z = P F(S) f(z). References [1] Colao, V., Marino, G. and Xu, H.-K., An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 2008, 344: 340-352. [2] Yao, Y.-H., A general iterative method for a finite family of nonexpansive mappings, Nonlinear Anal., 2007, 66: 2676-2687. [3] Maingé, P-E., The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, C. M. A., 2010, 59: 74-79. [4] Kurokawa, Y. Takahashi, W., Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal., 2010, 73: 1562-1568. [5] Yao, Y.-H., Noor, M. A. and Liou, Y.-C., On iterative methods for equilibrium problems, Nonlinear Anal., 2009, 70: 497-509. [6] Takahashi, W., N.-C., Wong and J.-C., Yao, Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems, Fixed Point Theory and Applications, 181, 2012. [7] Hu, C.-S. and Cai, K., Viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problem, Nonlinear Anal., 2010, 72: 1792-1808. [8] Li, H.-Y. and Su, Y.-F., Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems, Nonlinear Anal., 2010, 72: 847-855. [9] Qin, X.-L., Cho, S. Y. and Kang, S. M., Some result on generalized equilibrium problems involving a family of nonexpansive mappings, Applied Mathematics and Computation, 2010, 217: 3113-3126. [10] Ingarden, R.S., Über die Einbetting eines Finslerschen Rammes in einan Minkowskiischen Raum, Bull. Acad. Polon. Sci., 1954, 2: 305-308 (in German). [11] Zhang, D.-K., Zhou, H.-Y., Another iterative algorithm on fixed points for closed and quasi-nonexpansive mappings in Hilbert spaces, J. Hebei Normal Univ., 2009, 33: 579-581.